
Measurements (soft-)replacement as introduced in Deep Cascade [1]}: At each unrolled iteration, the values of the k-space of the current solution are (soft-)replaced by the original k-space measurements,
\(f_{\lambda }({\mathb {y}}, {\mathb {y}} _k, \Omega ) = (1 - M_{\Omega }) {\mathb {y}} _k + (1 - \lambda ) M_{\Omega } {\mathb {y}} + \lambda M_{\Omega } {\mathb {y}} _k \, ,\) 
with \(\lambda \in [0,1]\) . If \(\lambda = 0\) , then we have a full replacement.

Learned on the measurements space, as done for the PDNet [2]}, using convolutionnal networks:
\(f_{\theta }({\mathb {y}}, {\mathb {y}} _k, \Omega ) = CNN_{\theta }({\mathb {y}}, M_{\Omega } {\mathb {y}} _k)\, .\) 
In this case, \(f_{\theta }\)  is both the data consistency and the k-space correction module.

Measurements residual as done for the primal-only version of the PDNet and in the winning solution of the fastMRI challenge [3]}:
\(f({\mathb {y}}, {\mathb {y}} _k, \Omega ) = {\mathb {y}}- M_{\Omega } {\mathb {y}} _k \, .\)

